Link ¨oping Studies in Science and Technology Dissertations, No. 1756
Phase Noise and Wideband
Transmission in Massive
MIMO
Antonios Pitarokoilis
Division of Communication Systems Department of Electrical Engineering (ISY) Link ¨oping University, SE-581 83 Link ¨oping, Sweden
Phase Noise and Wideband Transmission in Massive MIMO
© 2016 Antonios Pitarokoilis, unless otherwise noted. ISBN 978-91-7685-791-5
ISSN 0345-7524
Abstract
In the last decades the world has experienced a massive growth in the de-mand for wireless services. The recent popularity of hand-held devices with data exchange capabilities over wireless networks, such as smart-phones and tablets, increased the wireless data traffic even further. This trend is not expected to cease in the foreseeable future. In fact, it is expected to accelerate as everyday apparatus unrelated with data communications, such as vehicles or household devices, are foreseen to be equipped with wireless communication capabilities.
Further, the next generation wireless networks should be designed such that they have increased spectral and energy efficiency, provide uniformly good service to all of the accommodated users and handle many more devices simultaneously. Massive multiple-input multiple-output (Massive MIMO) systems, also termed as large-scale MIMO, very large MIMO or full-dimension MIMO, have recently been proposed as a candidate technol-ogy for next generation wireless networks. In Massive MIMO, base stations (BSs) with a large number of antenna elements serve simultaneously only a few tens of single antenna, non-cooperative users. As the number of BS antennas grow large, the normalized channel vectors to the users become pairwise asymptotically orthogonal and, therefore, simple linear process-ing techniques are optimal. This is substantially different from the current design of contemporary cellular systems, where BSs are equipped with a few antennas and the optimal processing is complex. Consequently, the need for redesign of the communication protocols is apparent.
The deployment of Massive MIMO requires the use of many inexpensive and, potentially, off-the-shelf hardware components. Such components are likely to be of low quality and to introduce distortions to the information signal. Hence, Massive MIMO must be robust against the distortions intro-duced by the hardware impairments. Among the most important hardware impairments is phase noise, which is introduced by local oscillators (LOs)
at the BS and the user terminals. Phase noise is a phenomenon of particular importance since it acts multiplicatively on the desired signal and rotates it by some random and unknown argument. Further, the promised gains of Massive MIMO can be reaped by coherent combination of estimated chan-nel impulse responses at the BS antennas. Phase noise degrades the quality of the estimated channel impulse responses and impedes the coherent com-bination of the received waveforms.
In this dissertation, wideband transmission schemes and the effect of phase noise on Massive MIMO are studied. First, the use of a low-complexity single-carrier precoding scheme for the broadcast channel is investigated when the number of BS antennas is much larger than the number of served users. A rigorous, closed-form lower bound on the achievable sum-rate is derived and a scaling law on the potential radiated energy savings is stated. Further, the performance of the proposed scheme is compared with a sum-capacity upper bound and with a bound on the performance of the contem-porary multi-carrier orthogonal frequency division multiplexing (OFDM) transmission.
Second, the effect of phase noise on the achievable rate performance of a wideband Massive MIMO uplink with time-reversal maximum ratio com-bining (TR-MRC) receive processing is investigated. A rigorous lower bound on the achievable sum-rate is derived and a scaling law on the radi-ated energy efficiency is established. Two distinct LO configurations at the BS, i.e., the common LO (synchronous) operation and the independent LO (non-synchronous) operation, are analyzed and compared. It is concluded that the non-synchronous operation is preferable due to an averaging of the independent phase noise sources. Further, a progressive degradation of the achievable rate due to phase noise is observed. A similar study is extended to a flat fading uplink with zero-forcing (ZF) receiver at the BS. The fundamental limits of data detection in a phase-noise-impaired uplink are also studied, when the channel impulse responses are estimated via uplink training. The corresponding maximum likelihood (ML) detector is provided for the synchronous and non-synchronous operations and for a general parameterization of the phase noise statistics. The symbol error rate (SER) performance at the high signal-to-noise ratio (SNR) of the detectors is studied. Finally, rigorous lower bounds on the achievable rate of a Massive MIMO-OFDM uplink are derived and scaling laws on the radiated energy efficiency are stated.
Popul¨arvetenskaplig
sammanfattning
Det har skett en massiv tillv¨axt i efterfr˚agan p˚a tr˚adl ¨osa tj¨anster de senaste ˚artiondena. Dessutom blir handh˚allna enheter med f ¨orm˚aga att kommu-nicera ¨over tr˚adl ¨osa n¨atverk, som mobiltelefoner och surfplattor, allt van-ligare vilket g ¨or den trdl ¨osa trafiken ¨okar yttervan-ligare. Denna trend v¨antas inte upph ¨ora inom ¨oversk˚adlig framtid. I sj¨alva verket f ¨orv¨antas den ac-celerera i takt med att vardagliga apparater tidigare orelaterade till kom-munikation, s˚asom fordon eller hush˚allsapparater, planeras vara utrustade med tr˚adl ¨osa kommunikationsm ¨ojligheter. Stora leverant ¨orer inom den tr˚adl ¨osa kommunikationsindustrin f ¨orutsp˚ar en tiofaldig ¨okning av data-trafiken fram till ˚ar 2019 j¨amf ¨ort med 2014 och en tusenfaldig ¨okning ¨ar bara en tidsfr˚aga.
Vidare b ¨or n¨asta generation tr˚adl ¨osa n¨atverk utformas s˚a att de har ¨okad effektivitet i f ¨orh˚allande till den tillhandah˚allna datahastigheten och en-ergikonsumtionen, j¨amn kvalitet p˚a tj¨anster till anv¨andarna samt hantera m˚anga fler enheter samtidigt. Massiv MIMO (”input multiple-output”), ¨aven kallad hyper MIMO och storskalig MIMO, har nyligen f ¨oreslagits som en kandiderande teknik f ¨or n¨asta generations tr˚adl ¨osa n¨atverk. I massiv MIMO ¨ar basstationen utrustad med ett stort antal an-tal antenner och betj¨anar ett tioan-tal icke samarbetande,anv¨andare samtidigt. Med ¨okande antal antenner vid basstationen minskar st ¨orningar mellan kanaler till olika anv¨andare och enkla signalbehandlingstekniker har my-cket bra prestanda.
Utbyggnaden av massiv MIMO kr¨aver anv¨andning av m˚anga billiga och potentiellt sett icke specialgjorda komponenter. S˚adana komponenter kom-mer sannolikt att vara av l˚ag kvalitet och inf ¨ora f ¨orvr¨angningar av in-formationssignalen. D¨arf ¨or m˚aste Massiva MIMO vara robust mot dessa f ¨orvr¨angningar som kommer fr˚an anv¨andning av icke-ideal h˚ardvara.
En av de viktigaste funktionsneds¨attningarna fr˚an icke-ideal h˚ardvara ¨ar fasbrus. Denna funktionsneds¨attning introduceras av komponen-ter som kallas lokala oscillatorer som finns i basstationen s˚av¨al som i anv¨andarterminalerna. Fasbrus ¨ar ett fenomen av s¨arskild betydelse, efter-som den slumpvis roterar den ¨onskade signalen och kan inte ˚atg¨ardas genom att ¨oka effekten f ¨or den ¨onskade signalen. Vidare kan de utlo-vade vinsterna av massiv MIMO sk ¨ordas genom l¨amplig kombination av de mottagna signalerna vid basstationen. Emellertid f ¨orvr¨anger rotationen fr˚an fasbruset denna kombinering.
I denna avhandling studerar vi bredbands ¨overf ¨oringssystem och effekten av fasbruset p˚a Massive MIMO. I bredbands ¨overf ¨oring skiljer sig svaret fr˚an kanalen fr˚an en signal f ¨or varje frekvenskomponent. Detta resulterar i en f ¨orvr¨angning av den mottagna signalen och ett behov f ¨or en motta-garstruktur som har f ¨orm˚agan att ˚atervinna den s¨anda signalen med s˚a lite f ¨orvrngning som m ¨ojligt. I dagens system delas d¨arf ¨or bredband-skanalen upp i flera icke-st ¨orande smalbandiga kanaler, som d¨ampar sig-nalen j¨amnt. En teknisk term f ¨or denna typen av system ¨ar OFDM (”or-thogonal frequency-division multiplexing”). Men med massiv MIMO leder ansamlingen av ett stort antal oberoende slumpm¨assiga k¨allor ofta till en makroskopisk bild som verkar deterministisk. Detta kallas kanalh¨ardning. Vi unders ¨oker huruvida denna h¨ardande effekt f ¨orenklar den bearbetning som kr¨avs utan att beh ¨ova dela upp den bredbandiga kanalen i flera icke-interfererande smalbandiga kanaler, det vill s¨aga, utan att anv¨anda OFDM. Effekten av fasbrus i bredbandiga kanaler studeras ¨aven n¨ar en viss enkel linj¨ar behandlingstekniker anv¨ands. Eftersom det finns fler antennelement p˚a basstationen unders ¨oks det om det ¨ar mer f ¨ordelaktigt att anv¨anda en gemensam lokal oscillator eller flera oberoende lokala oscillatorer. Massiv MIMO kr¨aver sidoinformation om utbredningskanalen f ¨or att p˚a l¨amplig s¨att kombinera de mottagna signalerna vid basstationen. I massiv MIMO f ¨orv¨arvas denna sidoinformation med hj¨alp av f ¨orutbest¨amda signaler som kallas piloter. Fasbruset roterar dock den mottagna signalen vilket leder till en snedvridning av de mottagna piloterna. De grundl¨aggande gr¨anserna f ¨or kvaliteten p˚a sidoinformationen f ¨orv¨arvad via piloter i Massive MIMO, n¨ar fasbrus ¨ar n¨arvarande, unders ¨oks ocks˚a. Slutligen studeras effekten av fasbruset p˚a massiv MIMO med OFDM.
Acknowledgments
During the period of my doctoral studies there have been many people that played key role in my development. First and foremost, I would like to express my deepest gratitude to my main supervisor, Prof. Erik G. Larsson, for his support and supervision during all this period. His persistence to work on well-defined and important problems was pivotal to the quality and impact of my research. His meticulous review of the solutions that I was proposing was constantly transforming and improving my output and was helping me identify new directions and potential contributions. I am also heavily indebted to Dr. Saif Khan Mohammed, now Assistant Professor in IIT Delhi, who served as co-advisor for the first years of my doctoral studies. He has been tireless in explaining various problems of my proposed solutions and in clarifying fundamental, but difficult to grasp, notions from Information Theory. He has also been very persistent to teach me how I should express my ideas in my articles clearly and effectively. During the last years of my doctoral studies Associate Professor Dr. Emil Bj ¨ornson served as my co-advisor, to whom I have to extend my genuine appreciation for his help. His feedback has been immediate, concise and accurate in every issue that was coming up. His remarkable effectiveness helped me proceed forward and acquire a broader perspective of my re-search area. He has also assisted me to further improve my writing style. At the same time, I had the opportunity to work in very competitive envi-ronment with many other knowledgeable and helpful senior researchers and professors, such as Mikael Olofsson and Danyo Danev. A special thanks should be extended to Dr. Eleftherios Karipidis, now with Ericsson Research, who has helped me both with his technical knowledge and ex-pertise but also with his personal advice. I am also grateful to Prof. Michail Matthaiou, Queen’s University, Belfast, UK, for his continuing support and useful advice since my master’s studies in TU Munich. Among my doc-toral colleagues I would like to express my special gratitude to Dr. Hien
Quoc Ngo, with whom I have spent many hours discussing solutions on problems and clarifying the details of his work, which helped my research efforts. Finally, many thanks to all the other colleagues that have passed from the corridor of Communication Systems and have enlightened me with their presentations and during personal discussions.
Last but not least, I would like to thank my parents and my sister as silent and indirect “co-authors” of this dissertation. Their contribution was their support and care for all my life and the opportunity they gave me to work in a peaceful and comfortable environment free of distractions and worries.
Link ¨oping, April 2016 Antonios Pitarokoilis
Abbreviations
AGC Automatic Gain Control
AWGN Additive White Gaussian Noise
BER Bit Error Rate
BS Base Station
CC Constant Channel
CIR Channel Impulse Response
CSI Channel State Information
dB decibel
DFT Discrete Fourier Transform
DL Downlink
DMC Discrete Memoryless Channel
EVM Error Vector Magnitude
FC Fading Channel
FFT Fast Fourier Transform
FIR Finite Impulse Response
GBC Gaussian Broadcast Channel
IBI Inter-Block Interference
IEEE Institute of Electrical and Electronics Engineers i.i.d. Independent and Identically Distributed
ISF Impulse Sensitivity Function
ISI Inter-Symbol Interference
LLR Log-Likelihood Ratio
LMMSE Linear Minimum Mean Square Error
LO Local Oscillator
LoS Line-of-Sight
LPF Low-Pass Filter
LTI Linear Time Invariant
MAP Maximum A Posteriori
MIMO Multiple-Input Multiple-Output
ML Maximum Likelihood
MMSE Minimum Mean Square Error
MRC Maximum Ratio Combining
MRT Maximum Ratio Transmission
MU Multiuser
MUI Multiuser Interference
NS Non-Synchronous
OFDM Orthogonal Frequency Division Multiplexing OFDMA Orthogonal Frequency Division Multiple Access
PAM Pulse Amplitude Modulation
PAPR Peak-to-Average-Power-Ratio
PDP Power Delay Profile
PLL Phase-Lock Loop
PN Phase Noise
PSK Phase Shift Keying
QAM Quadrature Amplitude Modulation
QPSK Quadrature Phase Shift Keying
RF Radio Frequency
RZF Regularized Zero-Forcing
S Synchronous
SBS Symbol-by-Symbol
SER Symbol Error Rate
SI Side Information
SIMO Single-Input Multiple-Output
SINR Signal-to-Interference-plus-Noise Ratio SISO Single-Input Single-Output
SNR Signal-to-Noise Ratio
SU Single-User
TDD Time Division Duplex
TR Time-Reversal
UL Uplink
UNF Use-And-Forget
UT User Terminal
WLAN Wireless Local Area Network
ZF Zero-Forcing
Contents
Abstract v
Popul¨arvetenskaplig Sammanfattning (in Swedish) vii
Acknowledgments ix
Abbreviations xi
I Introduction 1
1 Motivation 3
2 Phase Noise in RF Oscillators 7
2.1 Macroscopic Manifestation of Phase Noise . . . 7
2.2 Phase Noise Sources . . . 9
2.3 Circuit-theoretic Modeling of Phase Noise . . . 11
2.3.1 The Model by Leeson [25] . . . 12
2.3.2 The Model by Hajimiri and Lee [13] . . . 13
2.3.3 The Model by Demir, et al. [14] . . . . 14
3 Phase Noise in Communication Systems 17 3.1 The Complex Baseband Representation in the Presence of Phase Deviations . . . 17
3.2 The AWGN Channel Impaired with Phase Noise . . . 19
3.3 The Band-Limited Phase Noise Channel . . . 21
3.4 The Wiener Phase Noise Model in Communications . . . 24
4 Massive MIMO with Phase Noise Impairments 27 4.1 The MU-MIMO Uplink Channel . . . 27
4.2 Massive MIMO . . . 29
4.3 Massive MIMO with Phase Noise . . . 31
4.3.1 Uplink Training . . . 32
4.3.2 Data Transmission . . . 32
4.4 Alternative Capacity Bound for Massive MIMO with Phase Noise . . . 35
5 Contributions of the Dissertation 41
5.1 Included Papers . . . 41
5.2 Not Included Papers . . . 45
5.3 Future Research Directions . . . 45
II Included Papers 59 A On the Optimality of Single-Carrier Transmission in Large-Scale Antenna Systems 61 1 Introduction . . . 64
2 System Model . . . 65
3 Achievable Sum-Rate . . . 66
4 Simulation Results . . . 71
B Uplink Performance of Time-Reversal MRC in Massive MIMO Systems Subject to Phase Noise 77 1 Introduction . . . 80
2 System Model . . . 82
2.1 Phase Noise Model . . . 83
2.2 Received Signal . . . 84
3 Transmission Scheme and Receive Processing . . . 84
3.1 Channel Estimation . . . 85
3.2 Time-Reversal Maximum Ratio Combining (TR-MRC) 86 4 Achievable Sum-Rate . . . 87
4.1 Exact Analysis of Synchronous versus Non-Synchronous Operation for a Toy Channel Model . . 94
5 Asymptotic Results . . . 95
6 Impact of Phase Noise Separately at the BS and at the User Terminals . . . 97
6.1 Special Case 1: Phase Noise Only at the UTs, σφ2 = 0 . 97 6.2 Special Case 2: Phase Noise Only at the BS, (σφ2 6= 0 and σθ2= 0) . . . 98
7 Numerical Examples . . . 99
8 Conclusions . . . 105
C Achievable Rates of ZF Receivers in Massive MIMO with Phase Noise Impairments 117 1 Introduction . . . 120
2 System Model . . . 121
3 Transmission Scheme and ZF Receiver . . . 122
3.1 LMMSE Channel Estimation . . . 123
3.2 Zero-Forcing (ZF) Equalization . . . 124
4 Achievable Rates . . . 125
5 Results - Discussion . . . 128
D ML Detection in Phase Noise Impaired SIMO Channels with Uplink Training 135 1 Introduction . . . 138 2 System Model . . . 141 2.1 Constant Channel (CC) . . . 142 2.2 Fading Channel (FC) . . . 143 3 Optimal Detectors . . . 144 3.1 Implementation of (15) and (25) . . . 148
3.2 The ML Detector for von Mises Phase Noise Increments148 4 High SNR Analysis . . . 150
4.1 High SNR Analysis for the Synchronous Operation . 150 4.2 High SNR Analysis for the CC-NS Case . . . 152
4.3 High SNR Analysis for the FC-NS Case . . . 154
5 Numerical Examples . . . 156
5.1 Extension to Longer Data Intervals . . . 161
6 Conclusions . . . 163
7 Two-Slot Proofs . . . 165
7.1 Proof of Proposition 1 for the CC-NS Case . . . 165
7.2 Proof of Proposition 1 for the FC-NS Case . . . 167
7.3 Proof of Proposition 2 for the CC-S Case . . . 169
7.4 Proof of Proposition 2 for the FC-S Case . . . 169
7.5 Proof of Proposition 3 . . . 169
8 T-slot Detectors . . . 170
8.1 Decision Feedback Detector: CC-NS . . . 170
8.2 “Better-than-optimal” Detector: CC-S . . . 172
8.3 Decision Feedback Detector: FC-NS . . . 173
8.4 “Better-than-optimal” Detector: FC-S . . . 175
8.5 Numerical Examples . . . 176
E Performance of the Massive MIMO Uplink with OFDM and Phase Noise 185 1 Introduction . . . 188 2 System Model . . . 189 3 Achievable Rates . . . 191 4 Numerical Examples . . . 194 5 Conclusion . . . 196
Part I
Chapter 1
Motivation
Arguably, the world has experienced a rapid growth of data traffic over wireless networks in the last decades. It is reported that the global mobile data traffic has increased by more than 400 million times over the past 15 years, from less than 10 GB per month in 2000 to 3.7 EB1per month at the end of 2015 [1]. This trend is not expected to cease any time soon. In fact it is predicted that it will accelerate in the years to come, as the traffic is expected to reach 30 EB per month by 2020 [1]. This demand is expected to originate not only from wireless data exchange by tablets and smartphones but also from the proliferation of new types of communication, such as machine-to-machine communication [2]. In short, any device that could collect and disseminate information via the wireless medium and could benefit from this data exchange, is expected to be equipped with wireless communica-tion capabilities.
The straightforward way to satisfy the increase in traffic demand is to in-crease the frequency spectrum for the communication. However, there are various reasons that make this approach not attractive. First, the spectrum is a constrained natural resource. Within a fixed spectrum portion multiple communication services, military and civilian, must be accommodated. Al-ready, most part of the available spectrum is allocated to various services and operators, so if we are to allocate more spectrum to mobile commu-nications, spectrum portions from other services must be made available. This might be possible, but only to a very restricted extent, which is def-initely not enough to fulfill the demand for the future mobile data traffic.
1
4 Chapter 1 Motivation
Also, certain frequency spectrum bands are not appropriate for wireless networks, due to unfavorable propagation conditions or high atmospheric attenuation. Finally, spectrum is a very expensive resource for mobile op-erators. It is clear that a more aggressive and spectrally efficient approach must be applied for the evolution of wireless networks. A more aggressive spectrum reuse can increase data rates within certain geographical areas of interest. This can be partly implemented with the deployment of small cells (including WiFi access points). However, small cells are less effective when dealing with high-mobility users and providing wide-area coverage. The deployment of multiple antennas at the transmitter and receiver (multiple-input multiple-output (MIMO)) was shown to provide significant gains in the amount information that can be communicated within a fixed frequency band. The gains scale linearly with the minimum of the trans-mit and receive antennas in the presence of rich scattering environment and when the channel is known at the receiver [3, 4]. Simultaneously with the emergence of point-to-point MIMO, the concept of multi-user MIMO (MU-MIMO) was investigated, where a base station (BS) equipped with a handful of antennas communicates with a few non-cooperative users over the same time and frequency resource [5]. Early work on the topic includes [6–8]. Point-to-point MIMO provide the promised gains in rich scattering scenarios, but require large antenna separation in the presence of a strong Line-of-Sight (LoS) component to achieve the same performance. In con-trast, MU-MIMO are more robust in LoS conditions. However, the adoption of MU-MIMO techniques in modern wireless communication standards has not been proportional to the available research literature due to the fact that the optimal transmission strategies are complex and require accurate chan-nel state information (CSI) at the BS.
Massive MIMO [9], also known as Large Scale Antenna Systems, Very Large MIMO, Full-Dimension MIMO, proposes a new approach towards the practical implementation of ideas from MU-MIMO, where a BS with an unprecedentedly large number of BS antenna elements, M , serves simul-taneously a few tens of single-antenna, non-cooperative users, K. When M ≫ K linear, low-complexity processing techniques are close to optimal, while at the same time, channel state acquisition can be made available at the BS via uplink training. This way the resources spent for channel train-ing are proportional to the number of users and, thus, the design is scalable with M . Massive MIMO is shown to achieve substantial gains in spectral and radiated energy efficiency of future wireless networks [10, 11].
5
Massive MIMO is not just an increase of the number of antenna elements at the BS and a corresponding adoption of linear processing strategies. This newly proposed physical layer can also have further implications on the de-sign of wireless networks. Currently, most wireless standards use multicar-rier techniques to transmit data over wireless channels. Even though multi-carrier transmission is very likely to be part of next generation wireless net-works, it would be useful to investigate whether with Massive MIMO there exist regimes or deployment scenarios where low complexity single-carrier transmission can provide comparable or even superior performance than currently used techniques. In particular, with single-carrier transmission low peak-to-average-ratio (PAPR) waveforms can be designed. With such waveforms, low quality, inexpensive, power efficient, non-linear power amplifiers can be used at each BS antenna element, which is important in Massive MIMO. In current systems expensive, power inefficient, highly lin-ear amplifiers with a large power back-off are required to transmit signals with high PAPR. Multi-carrier signals are known to have a large PAPR. With single-carrier transmission, all the available bandwidth is allocated to a user, which simplifies the user scheduling. Further, the transmission when M ≫ K will be robust against intersymbol interference. Finally, single-carrier transmission is less sensitive to impairments, such as carrier phase instabilities, in comparison to multi-carrier techniques, such as or-thogonal frequency division multiplexing (OFDM).
The deployment of economically viable Massive MIMO relies on the use of multiple inexpensive components that are likely to introduce distortions to the information signal. Hence, affordable Massive MIMO must be robust against hardware impairments that arise due to imperfections in the trans-mitter and the receiver circuits. Therefore, the study of the effect of hard-ware impairments, which is frequently treated as secondary in the study of conventional systems, is very important in Massive MIMO. One of the most important hardware impairments is phase noise, which is introduced in communication systems due to imperfections in the circuits of the local oscillators (LOs) at the transmitter and receiver chains. Imperfect LOs intro-duce random and time-varying rotations at the information signals. These rotations hinder the receiver to make the correct decisions about the trans-mitted signal based on the available noisy observations. Phase noise affects not only the data detection, but also the channel estimation. As Massive MIMO relies on the coherent combining of the received signals with esti-mated channel impulse responses in order to provide the promised gains, phase noise destroys the coherency between the channel estimates and the
6 Chapter 1 Motivation
effective channels at the time of decoding.
In this dissertation we investigate the performance of Massive MIMO with wideband transmission –single carrier and multi-carrier– and in the pres-ence of phase noise impairments. Capacity lower bounds are derived for various scenarios of practical interest and scaling laws with respect to ra-diated energy efficiency and other performance measures of interest are stated. The effect of phase noise on detection with training based CSI is also studied rigorously by deriving the maximum likelihood (ML) detec-tor. The introduction of the dissertation is structured as follows. In Chapter 2 the fundamental phenomena that give rise to phase noise in radio fre-quency oscillators are reviewed and the most important models are stated. In Chapter 3 the way that phase noise enters in communication systems is analyzed. In Chapter 4 the concept of Massive MIMO is briefly introduced and the basic steps for the analysis of phase noise impaired Massive MIMO systems are explained. In Chapter 5 the basic contributions of the disserta-tion are described and the included papers are listed.
Chapter 2
Phase Noise in RF Oscillators
In this chapter the issue of phase noise in radio frequency (RF) LOs is dis-cussed. The purpose is to relate the models used in this dissertation and in communications literature, in general, to well-known, established models that correspond to the physical reality of contemporary LOs. The discus-sion will be concentrated on free-running LOs, which is the main assump-tion on the oscillator operaassump-tion throughout the dissertaassump-tion. Initially, the macroscopic effects of phase noise on periodic waveforms are described. The main noise sources in electronic circuits that eventually perturb the waveform at the output of an LO are briefly reviewed. Finally, three main models that have been widely used to explain the behavior of phase noise are described.2.1
Macroscopic Manifestation of Phase Noise
The LOs are electronic circuits that are designed to produce an oscillat-ing waveform with a specified angular frequency, ωc. They are essential
in wireless communications, since they are used to modulate the baseband signal to passband at the transmitter and to demodulate the received pass-band signal to basepass-band at the receiver. Ideally, the output voltage, Vout(t),
of a noiseless LO is a sinusoidal waveform that is perfectly stable in ampli-tude, frequency and phase, given by
8 Chapter 2 Phase Noise in RF Oscillators
However, the electronic components (active and passive) that are used to manufacture the LO circuits are impaired by noise. Consequently, the out-put of a real LO has amplitude and phase that are time-varying and the output voltage is given by
Vout(t) = (1 + a(t)) cos (ωc(t + α(t))) , (2.2)
where a(t) is the amplitude perturbation and α(t) is the timing noise. Due to the amplitude limiting mechanism that is present in practical LOs, e.g., as an automatic gain control (AGC) component [12], the amplitude pertur-bation, a(t), gradually decays. Hence, the amplitude can be assumed to be constant. On the other hand, it has been shown that the noise α(t) persists and gives rise to a sequence of non-vanishing, random and varying timing offsets [13, 14].
In the time domain, phase noise becomes apparent by observing the zero-crossings of the oscillating waveform. The time interval between two con-secutive zero-crossings from positive to negative values in this oscillating waveform, which is usually the voltage or the current at the output of the LO, should be exactly equal to the period of the LO waveform. However, in the presence of phase noise, these zero-crossings are slightly shifted, as shown in Fig. 2.1a. In the frequency domain, the power spectral density of the LO output is a Dirac impulse located at ωc. In the presence of phase
noise, however, the power spectral density of the LO output, assuming that it is wide-sense stationary, widens around the oscillation angular frequency, ωc(Fig. 2.1b).
The effect of phase noise in digital modulation is a random phase rotation of the constellation points. In Fig. 2.2 the received symbols from a 16-QAM constellation [15] are plotted in the presence of phase noise when there is no additive thermal noise. The phase deviations in this figure are assumed to be zero-mean Gaussian1random variables with variance 10−2rad2/sample. It can be seen that the received symbols can be substantially rotated by phase noise. Consequently, the probability that a received signal is detected in error is high even when the desired signal power is significantly larger than the thermal noise power [16, 17]. The degradation due to phase noise is more detrimental for dense constellations, that convey information not only on the amplitude but also on the phase of the transmitted symbol. Phase noise can introduce distortion not only to the transmitted/received signal but also to signals in adjacent frequency bands. At the receiver
1
The justification of this choice is deferred to Section 2.3.3 and Section 3.4. 8
2.2. Phase Noise Sources 9 Samples 0 200 400 600 800 1000 Amplitude -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Noiseless Carrier Carrier with Phase Noise
(a) Phase noise in the time domain. The zero-crossings are shifted randomly from the nominal position in the presence of phase noise.
ωc
(b) Phase noise in the frequency domain. The ideal frequency spectrum widens due to phase noise.
Figure 2.1: Time-domain and frequency-domain manifestation of the out-put voltage from an LO with phase noise.
side, consider the demodulation of a weak passband signal with center fre-quency ω1, in the presence of a strong interferer in an adjacent frequency
band centered at ω2, as shown in Fig. 2.3a. In the case of a noisy LO,
sig-nificant interference can be introduced to the desired signal at ω1, from the
strong interferer at ω2, shown as the shaded spectrum in Fig. 2.3b. This
phenomenon is called reciprocal mixing [18]. At the transmitter side, a strong signal with a noisy LO at ω2, can introduce interference leakage to a weak
neighboring signal at ω1 (Fig. 2.4). Since the frequency bands around ω1
and ω2 are adjacent, the difference between the two carrier frequencies can
be of the order of a few kHz. This imposes strict requirements so that the power spectral density of the noisy LOs should decay very sharply [18].
2.2
Phase Noise Sources
Noise is unavoidable in electronic circuits. Various random microscopic events that occur within a circuit and its components give rise to fluctua-tions and disturbances of the current and the voltage at the output of the circuit. The random motion of electrons within a resistor result in a voltage,
10 Chapter 2 Phase Noise in RF Oscillators In-Phase -4 -2 0 2 4 Quadrature -4 -3 -2 -1 0 1 2 3 4 Scatter plot
Figure 2.2: 16-QAM constellation rotation in the presence of phase noise and the absence of additive thermal noise.
en(t), along the resistor that has a non-zero mean-square value even in the
absence of current flow through the resistor [19]. A resistor of R Ohm has a mean square noise density of
¯ e2
n= 4kT R∆f, (2.3)
over a bandwidth of ∆f Hz, where k ≈ 1.38 × 10−23Joule/Kelvin is the
Boltzmann constant and T is the temperature measured in Kelvin. This type of noise is referred to as thermal noise due to its dependence on the absolute temperature, T . The mean square noise density is independent of the frequency, f , for a very wide spectrum of frequencies, which implies that thermal noise is white for most of the frequencies of interest. Thermal noise is also known as Johnson-Nyquist noise, due to the scientists that first measured and explained the phenomenon [20, 21].
Shot noise appears in devices such as diodes or bipolar transistors in the presence of a constant average current flow of IDCAmp`ere. These devices
have a potential barrier that the charge carriers have to cross [19]. Due to the discrete type of the microscopic electron motions and the randomness in the time instants when these motions occur, fluctuations around IDCappear. A
shot noise source has a mean square current noise at a bandwidth of ∆f Hz given by
¯ i2
n= 2qIDC∆f, (2.4)
2.3. Circuit-theoretic Modeling of Phase Noise 11
ω1 ω2
(a) Weak desired signal at ω1, in the
presence of strong signal in an adja-cent frequency band adja-centered at ω2.
ω1 ω2
(b) Spectral widening due to noisy LO at the receiver. Strong interfer-ence from the adjacent signal at ω2,
leaks into the bandwidth of the de-sired signal at ω1.
Figure 2.3: Reciprocal mixing due to noisy LO at the receiver [18].
where q ≈ 1.6 × 10−19Coulomb is the electron charge. The mean square
current noise ¯i2
nis independent of the frequency, f , which implies that shot
noise is also white. The description and explanation of shot noise is at-tributed to W. Schottky [22].
Another type of noise that is present in physical systems is flicker noise. It has been observed in the fluctuations of various physical phenomena, span-ning from biology, to electronics and astrophysics. This type of noise is col-ored and has power spectral density that scales inverse-proportionally with the frequency. For this reason, flicker noise is also called 1/f -noise [23, 24]. The physical mechanism that generates flicker noise is still debatable in the physics community and its description involves empirical parameters [19].
2.3
Circuit-theoretic Modeling of Phase Noise
In this section we briefly review three of the most important models that have been used to explain the behavior of the output of noisy LOs. The models are presented in the chronological order that they appeared and the main contribution on the understanding and modeling of phase noise is concisely explained.
12 Chapter 2 Phase Noise in RF Oscillators
ω1 ω2
Figure 2.4: Interference leakage at the transmitter due to strong signal with high phase noise spectrum at ω2 in the bandwidth of the desired signal at
ω1.
2.3.1 The Model by Leeson [25]
The most common measure of the output of a noisy LO, usually a voltage or a current, is the single-sided power spectral density defined by [13]
L(∆ω)= 10 log∆ 10 P(ωc+ ∆ω, 1 Hz) Pc
, (2.5)
whereP(ωc + ∆ω, 1 Hz) is the single-sided power measured at frequency
band of width 1 Hz located at an offset of ∆ω from the carrier, ωc, and
Pc is the power of the carrier. The unit ofL(∆ω) is decibels below carrier
per Hertz, (dBc/Hz)2 [26]. The single-sided power spectral density at the
output of a noisy LO as a function of the offset, ∆ω, from the nominal os-cillation frequency, ωc, has, in general, a shape similar to the one shown in
Fig. 2.5. Three different regions are apparent, namely, i) the small offset region, where L(∆ω) drops as 1/f3, ii) the medium offset region, where
L(∆ω) drops as 1/f2and iii) the large offset region, whereL(∆ω) reaches a
constant level. Leeson in 1966 [25] attempted to explain this spectrum using linear time-invariant system theory. The derived spectrum is given by
L(∆ω) = 10 log10 " 2F kT Psig ( 1 + ωc 2Q∆ω 2) 1 +∆ω1/f3 |∆ω| # , (2.6)
2The careful reader will observe that the “per Hertz” should apply to the quantities
in-side the logarithm. However, in practice the unit is used as here, with the implicit under-standing that Hz should refer to the argument of the logarithm rather than the logarithm itself.
2.3. Circuit-theoretic Modeling of Phase Noise 13 L(∆ω) log(∆ω) ∆ω1/f3 2Qωc L(∆ω) = 10 log 2F kT Psig 1 + ωc 2Q∆ω 2 1 + ∆ω1/f 3 |∆ω|
Figure 2.5: Single-sided power spectral density of a noisy LO output as predicted by Leeson [25].
where ∆ω is the frequency offset from the carrier, k is the Boltzmann con-stant, T is the temperature in Kelvin, Psig is the signal power, Q is the
res-onator3, F is the device excess noise number [13] and ∆ω1/f3 is the
cor-ner frequency offset between the 1/f3 region and the 1/f2 region. Apart from matching the predicted spectrum at the output of a noisy LO, Leeson’s model also reveals some useful rules of thumb. For instance, an LO with high Q, i.e., good oscillator, will have a better phase noise performance. Leeson’s model, despite its heuristic nature, has been a major reference for many years.
2.3.2 The Model by Hajimiri and Lee [13]
Leeson’s model has certain drawbacks. For instance, F is an empirical pa-rameter that cannot be predicted before the design of the LO. Further, the value of ∆ω1/f3 in (2.6) must also be determined empirically in practice.
The reason for these deviations from reality is the fact that LOs are neither linear nor time-invariant systems. Hence, the Leeson approach, which re-lied on these assumptions, did not properly reflect the physical reality. Hajimiri and Lee [13, 26] showed that LOs are time-varying systems and dropped the time-invariance assumption. They represented the noise
3This is a parameter that shows the quality of the oscillator and is defined as the ratio of
14 Chapter 2 Phase Noise in RF Oscillators
sources in a general LO as current impulses and they argued that the re-sulting perturbations on the LO output can be decomposed to pure ampli-tude and pure phase perturbations. While the ampliampli-tude perturbations go to zero due to the amplitude limiting mechanisms in the circuit, the phase perturbations do not. Further, they argued that the impulse response of the varying phase, φ(t), at the LO output to a current noise impulse (current-to-phase transfer function) is still linear but time-varying. They introduced special functions that they called impulse sensitivity functions (ISFs), which contain information on the sensitivity of the output LO waveform to circuit noise impulses. In general, ISFs depend on the oscillating waveform and are usually determined numerically. The authors used the theory of linear time-varying systems to derive the spectrum at the output of a noisy LO. The derived model gives a more accurate and less heuristic explanation of the power spectral density of a noisy LO output. Therefore, it helps de-signers identify the dominant sources of phase noise and take them into consideration in their design.
2.3.3 The Model by Demir, et al. [14]
The models in Sections 2.3.1 and 2.3.2 explain the phenomenon of phase noise in terms of the output power spectral density rather than the time-domain statistical characterization of the phase noise process. Such a de-scription is more useful to the designers of LOs and less to communications engineers. An asymptotic statistical characterization of the phase noise pro-cess in the time domain is given by Demir, et al. [14]. They revisited the conjecture of Hajimiri and Lee that the perturbations can be decomposed to pure amplitude and pure phase perturbations and studied LOs as dynami-cal systems, i.e., systems that can be described by a differential equation in the presence of some perturbation
d
dtx(t) = f (x(t)) + B(x(t))b(t), (2.7) where x(t) is the state of the system–typically a vector of voltages and/or currents along capacitors and inductors of the LO–at time t, f (·) is some– not necessarily linear–function and B(x(t))b(t) is a small state-dependent perturbation. They assumed that there is a stable non-trivial periodic so-lution, xs(t), (“orbit” or “limit cycle” [27]) to the unperturbed system, i.e.,
(2.7) without the term B(x(t))b(t), and they showed that with a small per-turbation the orbit changes to xs(t + α(t)) + y(t), where y(t) is an
ampli-tude perturbation that eventually disappears, whereas the time shift, α(t), 14
2.3. Circuit-theoretic Modeling of Phase Noise 15
persists. Hence, they showed that the behavior of α(t) demonstrates the accumulative nature of phase noise. They also proceeded to show that, asymptotically in t and when the circuit noise sources are white (thermal and shot noise), the characteristic function of α(t) is that of a Gaussian ran-dom variable and that the autocorrelation function of the time shift α(t) is given by
E[α(t)α(t + τ )] = m2+ c min(t, t + τ ), (2.8) where m and c are constants. The asymptotic statistical characterization of α(t) in (2.8) implies that the increments α(t)− α(t + τ) are Gaussian and have variance that is proportional to|τ| [28]. These properties correspond to a continuous-time Brownian motion or Wiener process [29, Section 37] and jus-tify a model that is widely used in communications, the Wiener phase noise model. The fact that phase noise can be described in the time domain with a very compact characterization that depends on a single scalar, c, makes the Wiener model very attractive for study of phase noise in the fields of Information Theory and Detection & Estimation Theory. A more detailed analysis of this model in a communications setup is deferred to Section 3.4. Demir extended the analysis to include white and colored noise sources (i.e., flicker noise), however, the expressions attained are much more com-plicated and involve the integration over the spectra of the colored noise sources [30, Lemma 7.1].
16 Chapter 2 Phase Noise in RF Oscillators
Chapter 3
Phase Noise in Communication
Systems
In the following the impairment of phase noise is introduced in suitable communication system models. The discussion starts with the complex baseband representation of a real passband signal as it passes through a communication system with phase deviations at the transmitter and the receiver. Subsequently, the single-input single-output (SISO) phase noise impaired additive white Gaussian noise (AWGN) channel is discussed and based on the continuous-time system model, a discrete-time approximation is derived. The band-limited SISO channel with phase noise impairments at the transmitter and the receiver is also introduced and discrete-time ap-proximations are derived. The chapter concludes with the description of the Wiener phase noise model, which has been motivated by Section 2.3.3, and its connection to the communication-theoretic system models is de-scribed.
3.1
The Complex Baseband Representation in the
Presence of Phase Deviations
Let two sequences a(t) and b(t) of pulse amplitude modulation (PAM) sym-bols to be transmitted over a channel located at a frequency, fc. The channel
18 Chapter 3 Phase Noise in Communication Systems a(t) b(t) c(t) d(t) √ 2 cos(2πfct + φTX(t)) √ 2 cos(2πfct + φRX(t)) −√2 sin(2πfct + φTX(t)) − √ 2 sin(2πfct + φRX(t)) ˜ s(t)
Figure 3.1: Passband representation of a system with phase deviations at the transmitter and the receiver.
the transmitted signal. However, we assume that phase noise distortions, φTX(t) and φRX(t), are introduced due to noisy LOs at the transmitter and
the receiver, respectively. The real passband signal after the modulation is given by ˜ s(t) = a(t)√2 cos(2πfct + φTX(t))− b(t) √ 2 sin(2πfct + φTX(t)) =ℜn(a(t) + jb(t))ejφTX(t)√2ej2πfcto. (3.1)
Hence, the complex baseband equivalent representation of the transmitted signal in the presence of phase deviations at the transmitter is given by sb(t) = (a(t) + jb(t))ejφTX(t).
At the receiver, the in-phase component, c(t), is given by c(t) = LPFns(t)˜ √2 cos(2πfct + φRX(t))
o
= a(t) cos(φTX(t)− φRX(t))− b(t) sin(φTX(t)− φRX(t)), (3.2)
where LPF{·} is the operation of a low-pass filter, which filters out the fre-quency component at 2fc. Similarly, the quadrature component is given
by
d(t) = LPFns(t)˜ √2 sin(2πfct + φRX(t))
o
= a(t) sin(φTX(t)− φRX(t)) + b(t) cos(φTX(t)− φRX(t)). (3.3)
3.2. The AWGN Channel Impaired with Phase Noise 19
The received complex baseband signal is
y(t) = c(t) + jd(t) = (a(t) + jb(t))ejφTX(t)e−jφRX(t). (3.4)
In the case of narrowband SISO systems, the baseband equivalent represen-tation in (3.4) reduces to y(t) = (a(t) + jb(t))ejθ(t), where only the difference θ(t) = φTX(t)− φRX(t) is important. However, in MU-MIMO systems the
distinction between the transmit and receive phase deviations is essential. Consider for example a system with two non-cooperative single-antenna transmitters and a single-antenna receiver. The two effective channels from the transmitters to the receiver will be both affected by the same receive phase noise process but from independent transmit phase noise processes. This must be reflected by the selected channel representation.
3.2
The AWGN Channel Impaired with Phase Noise
xν x(t) s(t) z(t) y(t) yi gT(t) gR(t) ejθT(t) n(t)˜ e−jθR(t) t = iTs
Figure 3.2: The continuous-time additive white Gaussian (AWGN) channel with phase noise.
Based on the complex baseband equivalent representation of phase noise impaired SISO systems from Section 3.1, the phase noise impaired AWGN channel is studied in continuous time and a discrete time approximation is derived. Let the waveform to be transmitted, x(t), be
x(t) =
N
X
ν=1
xνgT (t− νTs) , (3.5)
where Tsis the symbol interval,{x1, . . . , xN} is the sequence of transmitted
symbols selected from a fixed constellation [15] and gT(t) is the impulse
re-sponse of the transmit pulse shaping filter. The actually transmitted wave-form is disturbed by the transmit phase noise process, θT(t):
s(t) = ejθT(t)x(t) = ejθT(t)
N
X
ν=1
20 Chapter 3 Phase Noise in Communication Systems
The channel introduces to the transmitted signal, s(t), additive noise ˜n(t), which is a white Gaussian random process. At the receiver the signal after the downconversion by the LO at the receiver, z(t), is given by
z(t) =
N
X
ν=1
xνe−j(θR(t)−θT(t))gT(t− νTs) + n(t). (3.7)
Due to the circular symmetry of the ˜n(t) noise process, n(t) = e−jθR(t)n(t)˜
and ˜n(t) have the same second order statistical characterization. Further, it is clear that θT(t) and θR(t) are observed only through their difference.
Hence, in what follows we note θ(t)= θ∆ T(t)−θR(t). At the receiver, the
sig-nal is filtered by a linear time-invariant (LTI) filter with impulse response, gR(t), which is matched to gT(t). The output of the filter gR(t) is given by
y(t) = Z +∞ −∞ gR(τ )z(t− τ)dτ = N X ν=1 xν Z +∞ −∞ ejθ(t−τ )gT∗(−τ)gT (t− τ − νTs) dτ + Z +∞ −∞ gR(τ )n(t− τ)dτ. (3.8)
The mathematical manipulation of the derived continuous-time model ap-pears to be formidable. In addition, the progress of digital electronics mo-tivates the discretization of the continuous-time model in (3.8). For this purpose, the received signal, y(t), is sampled at regular intervals equal to the symbol interval, Ts. The sample yiat time t = iTsis given by
yi= y(t = iTs) = N X ν=1 xν Z +∞ −∞ ejθ(iTs−τ )g∗ T(−τ)gT (iTs− τ − νTs) dτ + ni (3.9)
A common assumption is that phase noise is constant within a symbol in-terval, but it does change between two consecutive symbol intervals. Under this assumption and if the cascade of the transmit and receive filters gT(t)
and gR(t) satisfies the Nyquist criterion for intersymbol interference [15]
the following simple symbol-sampled phase noise channel model follows immediately
yi= xiejθi+ ni, (3.10)
where the noise samples ni are jointly Gaussian and uncorrelated, hence
statistically independent [31, Section 3.3.3]. Under these conditions, the se-quence y1, . . . , yN is sufficient statistics for the detection of x1, . . . , xN. The
3.3. The Band-Limited Phase Noise Channel 21
model (3.10) has been extensively studied under various assumptions on the statistics of the phase sequence,{θi}. The arguments used above for the
derivation of the discrete-time symbol-sampled phase noise model are in general well-known and can be found in literature either implicitly [32] or more explicitly [33, 34].
3.3
The Band-Limited Phase Noise Channel
xν x(t) s(t) z(t) y(t) c(t) yi gT(t) gR(t) ejθT(t) n(t)˜ e−jθR(t) t = iTs
Figure 3.3: The continuous-time band-limited channel with phase noise. Here we extend the discussion to the case of a band-limited channel with complex baseband equivalent impulse response c(t), when phase noise is introduced at the transmitter and the receiver, as shown in Fig. 3.3. We assume that the signal to be transmitted is given by
x(t) =
N
X
ν=1
xνgT (t− νTs) .
The transmitter signal, which is distorted by transmit phase noise, is given by s(t) = ejθT(t)x(t) = ejθT(t) N X ν=1 xνgT(t− νTs) . (3.11)
The received signal after the filtering from the channel with impulse re-sponse c(t), the distortion by additive Gaussian noise, ˜n(t), and the de-modulation by the imperfect LO at the receiver is given by
z(t) = e−jθR(t) Z +∞ −∞ c(τ1)s(t− τ1)dτ1+ e−jθR(t)n(t)˜ = e−jθR(t) Z +∞ −∞ c(τ1) ejθT(t−τ1) N X ν=1 xνgT (t− τ1− νTs) ! dτ1+ e−jθR(t)˜n(t) = N X ν=1 xν Z +∞ −∞ e−j(θR(t)−θT(t−τ1))c(τ 1)gT (t− τ1− νTs) dτ1+ n(t)
22 Chapter 3 Phase Noise in Communication Systems
At the output of the receive filter, gR(t), the signal is given by
y(t) = Z +∞ −∞ gR(τ2)z(t− τ2)dτ2 = N X ν=1 xν Z +∞ −∞ Z +∞ −∞ e−j(θR(t−τ2)−θT(t−τ2−τ1)) (3.12) × gR(τ2)c(τ1)gT (t− τ2− τ1− νTs) dτ1dτ2+ Z +∞ −∞ gR(τ2)n(t− τ2)dτ2
It is clear that a discrete-time model derived from the continuous-time representation is necessary. A rigorous argumentation that can lead to a tractable discrete-time description appears to be difficult. Often an approx-imation similar to the one applied to discretize (3.9) is invoked. That is, the phase noise processes are assumed to be significantly more narrowband than the phase-noise-free overall pulse shape h(t) = (gR∗ c ∗ gT)(t). Under
this assumption, the transmit and receive phase noise processes are treated as one and the discrete-time model is given by
yi= e−jθ[i] L−1
X
l=0
hlxi−l+ ni, (3.13)
where hl = (gR∗ c ∗ gT)(lTs)1and θ[i] is the sampled equivalent phase noise
process. Such an approach has been adopted by many authors, such as [28, 35–37], which are studies of single-input single-output (SISO) orthogonal frequency division multiplexing (OFDM) systems.
There are two outstanding issues with the discrete-time model in (3.13). First, the phase noise processes at the transmitter and the receiver are treated as an equivalent phase noise process. This is a problem in MIMO and multi-user systems, where independent phase noise processes at differ-ent antenna elemdiffer-ents or at differdiffer-ent users must be appear separately. Sec-ond, the phase noise processes at the transmitter and the receiver, θT(t) and
θR(t), in (3.12) are distinguishable as ejθT(t) is convolved with the
propa-gation channel, c(t), as well as with the receive filter, gR(t). In contrast,
e−jθR(t) is only convolved with g
R(t). We seek for a discrete-time
approxi-mation that will reflect this property of the continuous-time representation.
1
Here, it has also been assumed that the overall pulse shape, h(t), gradually decays to negligible values, so we can truncate the sampled impulse response to L symbol-spaced channel taps.
3.3. The Band-Limited Phase Noise Channel 23
We assume that the transmit and receive filters, gT(·) and gR(·), are ideal
integrate-and-dump filters, i.e., their impulse responses are given by
gT(t) = gR(t) =
( 1
√
Ts, 0≤ t ≤ Ts
0, elsewhere. (3.14)
Then the continuous-time model in (3.12) is expressed as y(t) = N X ν=1 xν Z Ts 0 Z t−τ2−νTs t−τ2−νTs−Ts e−j(θR(t−τ2)−θT(t−τ2−τ1)) 1 Ts c(τ1)dτ1dτ2 + Z Ts 0 1 √ Ts n(t− τ2)dτ2 (3.15)
With equidistant symbol-spaced sampling at t = iTs, the discrete-time
model derived from the continuous-time model in (3.15) is given by yi = y(t = iTs) = N X ν=1 xl Z Ts 0 Z (i−ν)Ts−τ2 (i−ν−1)Ts−τ2 e−j(θR(iTs−τ2)−θT(iTs−τ2−τ1)) 1 Ts c(τ1)dτ1dτ2 + Z Ts 0 1 √ Ts n(iTs− τ2)dτ2. (3.16)
We invoke the piecewise-continuous assumption on the phase noise pro-cesses successively on θT(·) and θR(·), i.e., we assume that the phase noise
process has not changed substantially within an interval of duration Ts, and
we also define ˜ c(t)=∆ 1 Tc Z t t−Ts c(τ )dτ. (3.17) and ni=∆ Z Ts 0 1 √ Ts n(iTs− τ)dτ. (3.18)
Then, the discrete-time approximation is given by yi ≈ N X ν=1 xνejθT(νTs) Z Ts 0 e−jθR(iTs−τ2)˜c((i− ν)T s− τ2)dτ2+ ni ≈ N X ν=1 xνe−jθR((i−1)Ts)ejθT(νTs) Z Ts 0 ˜ c((i− ν)Ts− τ2)dτ2+ ni.
24 Chapter 3 Phase Noise in Communication Systems
With the additional definition and truncation of the discrete-time impulse response
hl =∆
(RTs
0 ˜c(lTs− τ)dτ, for 0≤ l ≤ L − 1
0, elsewhere (3.19)
a discrete-time approximation of the continuous-time model in (3.12) is given by
yi= e−jθR[i] L−1X
l=0
hlejθT[i−l]xi−l+ ni. (3.20)
The system model in (3.20) is appealing since the phase noise processes at the transmitter and the receiver appear explicitly and the convolution effect of the propagation channel on θT(t) is also given as a discrete-time
convo-lutional sum. Consequently, it has been extensively used in the literature to model phase noise in frequency selective channels [38–40]. However, in contrast to the discrete-time model in (3.10), a detailed derivation of the model (3.20), as the one given here, appears to be absent in the relevant literature.
3.4
The Wiener Phase Noise Model in
Communica-tions
In the Sections 3.1, 3.2 and 3.3 we described the way that phase noise ap-pears in standard communication system models [38, 41–44]. In this section we present a widely studied model on phase noise in the fields of Informa-tion Theory and DetecInforma-tion & EstimaInforma-tion Theory. Based on the discussion of Section 2.3.3, the phase noise process, θ(t) = ωcα(t), in (3.8) is modeled as a
continuous-time Wiener process, i.e., θ(t)− θ(t0) =
Z t t0
w(τ )dτ, (3.21)
where t0is the start of the observation of the process and w(t) is a real white
Gaussian process with E [w(t)] = 0 and E [w(t1)w(t2)] = ωc2cδ(t1 − t2). Of
particular interest is the autocorrelation and power spectral density of the
3.4. The Wiener Phase Noise Model in Communications 25
process ζ(t) = ej(θ(t)−θ(t0)). It is straightforward from the definitions of ζ(t)
and θ(t) to show that
Rζ(τ ) = E [ζ(t)ζ∗(t + τ )] = E
h
ej(θ(t)−θ(t0))e−j(θ(t+τ )−θ(t0))i
= Ehejωc(α(t)−α(t+τ ))i= e−ω22c c|τ |. (3.22)
The fact that Rζ(τ ) is only function of the relative time lag τ implies the
stationarity of the process ζ(t). The corresponding power spectral density is Sζ(f ) = Z +∞ −∞ Rζ(τ )e−j2πfτdτ = Z +∞ −∞ e−ω2c c2 |τ |e−j2πfτdτ = 1 π πfc2c π2f4 cc2+ f2 = 1 π β/2 (β/2)2+ f2. (3.23)
The shape of the spectrum in (3.23) is Lorentzian and β = 2πf∆ c2c is the double-sided 3 dB bandwidth of the process. Observe that this shape is an accurate description of the medium offset region of L(∆ω) in Fig. 2.5. However, it does not exhibit the 1/f3behavior for small offsets from carrier. This is due to the fact that the Wiener phase noise characterization is valid for LOs with only white circuit noise sources (thermal and shot noise) and the 1/f3 behavior is attributed to colored circuit noise sources, e.g., flicker noise.
In the discrete time, the continuous-time Wiener process is sampled at a frequency 1/Tsto yield the discrete-time Wiener process, where
θi+1= θi+ wi, (3.24)
where θi = θ(iTs) and
wi =
Z (i+1)Ts
iTs
w(τ )dτ.
From the properties of the continuous-time Wiener process, the increments wi are independent identically distributed (i.i.d.) zero mean real Gaussian
random variables with variance σw2 = 2πβTs, i.e., wi ∼ NR(0, σw2). In
information-theoretic studies it is often assumed that the initial phase θ0
is uniform in [0, 2π), i.e., θ0 ∼ U[0, 2π), and then the process {θi} is
26 Chapter 3 Phase Noise in Communication Systems
Chapter 4
Massive MIMO with Phase
Noise Impairments
In this chapter, the concept of MU-MIMO systems is introduced and the basic system model is described for the single-cell case. The introduction of Massive MIMO follows, where the main differences with conventional MU-MIMO systems are explained. The outline of the achievable sum-rate analysis in Massive MIMO is briefly presented. Subsequently, phase noise impairments are introduced into the basic Massive MIMO model. Through simple yet illustrative examples, the fundamental differences in the achiev-able sum-rate analysis between the phase noise case and phase-noise-free case are reviewed. The capacity lower bounds used in this dissertation are presented in detail, their desirable characteristics are mentioned and the potential use of alternative capacity lower bounds is discussed.
4.1
The MU-MIMO Uplink Channel
Consider a BS with M antenna elements serving simultaneously K single-antenna, non-cooperative users1within a geographical area, called cell. We study the direction of communication where the users send data to the BS; this is called uplink. Narrowband transmission is assumed, i.e., the effect
1The single-antenna per user assumption is not essential and is done here for simplicity.
28 Chapter 4 Massive MIMO with Phase Noise Impairments
of the propagation through the channel between a user and a BS antenna is represented by a single complex scalar coefficient. We denote the channel gain from the k-th user to the m-th BS antenna by hm,k and the channel
vector from the k-th user to the BS by hk ∆
= [h1,k, . . . , hM,k]T and assume
that it is a circularly symmetric complex Gaussian random vector, i.e., hk ∼
NC(0, IM). If the k-th user transmits the symbol xk, which is subject to an
average power constraint, E|xk|2 ≤ 1, then the received vector at the BS
is given by y=√ρ K X k=1 hkxk+ wk, (4.1)
where y ∈ CM is an M -dimensional complex vector and wk is additive
white Gaussian noise, distributed as wk ∼ NC(0, IM). If E|hm,k|2 =
1, ∀m, k and only one user transmits, then ρ is the expected signal-to-noise-ratio (SNR) at each BS antenna. Important research results on the limits of communication on the uplink channel are already available in the litera-ture [46–49]. The downlink, which is the communication direction from the BS to the users, is also important and has been extensively studied [50–57]. The concept of MU-MIMO systems was in itself a paradigm shift with re-spect to point-to-point single-user (SU) MIMO. In point-to-point SU-MIMO both the BS and the user terminal are equipped with multiple antenna ele-ments and different users are scheduled in orthogonal channels. In the pres-ence of a strong LoS path between the user and the BS or of strong spatial correlation, the number of streams that can be multiplexed in the SU-MIMO case can drop to one, offering only a small performance gain in comparison to the case where the BS and the user have a single antenna. However, MU-MIMO is more robust under LoS conditions, since two users are likely to be spatially separated and cases of strong correlation among users can be resolved by appropriate multi-user scheduling. Also, due to the single-antenna assumption for the user terminals, the complexity is moved to the BS and the user terminals can be small and energy efficient. However, MU-MIMO has also drawbacks. Accurate knowledge of the channels, hk, is
re-quired to reap the gains of MU-MIMO. This knowledge acquisition is costly, particularly in the downlink direction [5]. In addition, the optimal transmit and receive strategies in MU-MIMO, where the number of BS antennas, M , is approximately equal to the number of single-antenna users, K, are non-linear with very high complexity and simpler non-linear schemes often perform poorly.
4.2. Massive MIMO 29
4.2
Massive MIMO
Massive MIMO proposes a further shift in the paradigm of cellular systems design [58, 59]. A Massive MIMO BS is equipped with an unprecedentedly large number of antenna elements, M , a few hundreds or even thousands, and serves simultaneously a few tens of non-cooperative, single-antenna users, K. In this operational regime, where M ≫ K, the normalized propa-gation vector channels from different users tend to become asymptotically orthogonal. This is called favorable propagation. Further, the normalized channel norms tend to be very close to their statistical mean. This phe-nomenon is called channel hardening. With favorable propagation and chan-nel hardening, linear transmit and receive strategies are close to optimal. Similar to conventional MU-MIMO, accurate knowledge of the channels at the BS side is required in Massive MIMO in order to reap the promised gains. This knowledge is typically acquired via uplink training. With this choice, the length of the required training interval increases with the num-ber of terminals and not with the numnum-ber of BS antennas. Finally, the BS can use the channel knowledge acquired via uplink training to detect the signals received in the uplink.
The concept of coherence interval is key in the study of wireless channels, and of Massive MIMO in particular. A coherence interval is the block, of say τcchannel uses, during which the channel remains approximately
con-stant. In the present exposition of Massive MIMO the coherence interval is split into an uplink training interval of τp ≥ K channel uses and an
up-link data interval of τd= τc − τp channel uses2. During training, each user
transmits a deterministic training sequence √τpψkthat is orthogonal to the
training sequences of the other users, i.e., ψkHψk′ = δ(k− k′), where δ(·) is
the Kronecker delta. The received vectors during training are given by
Yp = √ρτp K X k=1 hkψTk + Wp (4.2) Ypψ∗k= √ρτphk+ Wpψ∗k, (4.3)
where Yp= [y[1], . . . , y[τp]] and Wp = [w[1], . . . , w[τp]]. The BS forms an
es-timate, ˆhk, for the channel vector, hk, based on Ypψ∗kin order to detect the
2
In Massive MIMO part of the coherence interval is allocated also for downlink data transmission and, possibly, for downlink pilots [60], but in this introductory exposition the focus is on the uplink.
30 Chapter 4 Massive MIMO with Phase Noise Impairments
information symbols transmitted from the k-th user, xk[τp + 1], . . . , xk[τc].
Information symbols from different users are assumed to be independent. During the data transmission interval, the BS uses the estimated channels3
with maximum ratio combining (MRC)4to detect the transmitted data sym-bol, xk[i], at the i-th channel use of the data interval, i.e.,
ˆ xk[i] = ˆhHky[i] =√ρ K X k′=1 ˆ hHkhk′xk′[i] + ˆhHkw[i]. (4.4) If Ehx∗khˆHkw[i] hˆk i = 0 and EhhˆHk hkxk ∗ ˆ hHkw[i] hˆk i = 0, a lower bound (achievable rate), RSIk, on the maximum mutual information between xk[i]
and ˆxk[i] conditioned on ˆhk, I(xk[i]; ˆxk[i]|ˆhk) is obtained by
max
pXk[i](xk[i])
I(xk[i]; ˆxk[i]|ˆhk)≥ RSIk (4.5)
= E log2 1 + ρ E h ˆ hHkhk hˆk i 2 ρPKk′=1E hˆHkhk′ 2 ˆ hk − ρ E h ˆ hHkhk ˆhk i 2 + hˆk 2 .
The maximization is over all the input densities pXk[i](xk[i]) that satisfy the
power constraint Eh|xk[i]|2
i
≤ 1. We will use the term side-information bound for RSIsince it explicitly uses the side information acquired via uplink
training both to process the received signal vector y[i] and to decode the information symbol xk[i] from the output, ˆxk[i], of the receive processing
filter. A more tractable but looser bound, RUNFk , on the maximum mutual information between xk[i] and ˆxk[i], I(xk[i]; ˆxk[i]), is given by
max
pXk[i](xk[i])
I(xk[i]; ˆxk[i])≥ RUNFk (4.6)
= log2 1 + ρ E h ˆ hHkhk i 2 ρPKk′=1E hˆHkhk′ 2 − ρ E h ˆ hHkhk i 2 + E hˆk 2 .
We will use the term use-and-forget bound for RUNFsince ˆhk is only used to
process the received vector y[i] but this information is not explicitly used to
3The users have no knowledge of the estimates ˆh
k, hence the data symbols xk[i] are
independent of the estimates ˆhk.
4Some other linear or with low complexity processing scheme, such as linear minimum
mean square error (LMMSE), is also possible. 30
4.3. Massive MIMO with Phase Noise 31
decode the information symbol xk[i] from ˆxk[i]. A detailed derivation and
information-theoretic justification of these bounds can be found in [61]. We note that these bounds first appeared in [62–64] and since then they have been extensively used in the study of Massive MIMO systems [10,11,65–67].
4.3
Massive MIMO with Phase Noise
In this section the basic single-cell system model in (4.1) is augmented with the effect of transmit and receive phase noise and the methodology for the derivation of achievable rates is outlined and explained. For simplicity, the exposition here is restricted to a single user, however, all the fundamental phenomena that appear due to phase noise are readily revealed. The treat-ment of more complicated models can be found in the papers that follow. At the i-th channel use of the coherence interval, the received signal at the m-th BS antenna is given by
ym[i] =√ρe−jθm[i]hmejφ[i]x[i] + wm[i], (4.7)
based on the analysis leading to (3.20) (with L = 1). The processes θm[·],
φ[·], hm, x[·] and wm[·] are mutually independent. However, BS phase noise
processes from the m1-th and m2-th BS antennas, θm1[·] and θm2[·],
respec-tively, can be arbitrarily dependent. We consider two particular operations; the synchronous operation where θ1[·] ≡ · · · ≡ θM[·] and the non-synchronous
operation where the θm[·] are mutually independent along the BS antennas.
The synchronous operation models a centralized deployment where one LO provides the carrier waveform to all the BS antennas. In contrast, the non-synchronous operation models a distributed deployment where sepa-rate LOs are used for every BS antenna. All the phase noise processes are assumed to be discrete-time Wiener processes as defined in Section 3.4. The variances of the phase noise increments for θm[·] and φ[·] are σ2θ and σ2φ,
respectively. The matrix-vector formulation is given by
y[i] =√ρΘ[i]hejφ[i]x[i] + w[i], (4.8) where y[i] ∈ CM, Θ[i] = diag∆ e−jθ1[i], . . . , e−jθM[i] for the
non-synchronous operation and Θ[i]= e∆ −jθ[i]IM for the synchronous operation.
As it has been outlined in Section 4.2, a coherence interval of τc channel
